PoS(LAT2009)104
maximally twisted mass fermions
R. Barona, B. Blossierb, P. Boucaudb, A. Deuzemanc, V. Drachd, F. Farchionie, V. Gimenezf, G. Herdoizag, K. Janseng, C. Michaelh, I. Montvayi, D. Palaof, E. Pallantec, O. Pèneb, S. Reker∗†c‡, C. Urbachj, M. Wagnerkand U. Wengerl
aCEA, Centre de Saclay, IRFU/Service de Physique Nucléaire, F-91191 Gif-sur-Yvette, France
bLaboratoire de Physique Théorique (Bât. 210), Université de Paris XI, Centre d’Orsay, 91405 Orsay-Cedex, France
cCentre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands
dLaboratoire de Physique Subatomique et Cosmologie, 53 avenue des Martyrs, 38026 Grenoble, France
eUniversität Münster, Institut für Theoretische Physik, Wilhelm-Klemm-Straße 9, D-48149 Münster, Germany
fDep. de Física Teòrica and IFIC, Universitat de València-CSIC, Dr.Moliner 50, E-46100 Burjassot, Spain
gNIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
hDivision of Theoretical Physics, University of Liverpool, L69 3BX Liverpool, United Kingdom
iDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany
jHelmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany
kHumboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, Germany
lAlbert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland
We present first results from runs performed withNf =2+1+1 flavours of dynamical twisted mass fermions at maximal twist: a degenerate light doublet and a mass split heavy doublet. An overview of the input parameters and tuning status of our ensembles is given, together with a comparison with results obtained withNf=2 flavours. The problem of extracting the mass of the K- andD-mesons is discussed, and the tuning of the strange and charm quark masses examined.
Finally we compare two methods of extracting the lattice spacings to check the consistency of our data and we present some first results ofχPT fits in the light meson sector.
The XXVII International Symposium on Lattice Field Theory - LAT2009 July 26-31 2009
Peking University, Beijing, China
∗Speaker.
†For the ETM Collaboration
‡Email: s.f.reker@rug.nl. Preprint numbers: DESY 09-175, HU-EP-09/50, MS-TP-09-22, SFB/CPP-09-98
PoS(LAT2009)104
1. Introduction
The twisted mass formulation of Lattice QCD [1,2] is being studied extensively withNf =2 dynamical flavours by the European Twisted Mass (ETM) collaboration [3–7]. In this formulation of QCD, the Wilson term is chirally rotated within an isospin doublet. To include a dynamical strange quark in a unitary setup, we add, in addition to the strange quark a charm quark in a heavier and mass-split doublet as discussed in [8–10]. We will briefly describe our action in section 2, recapitulate our procedure for tuning to maximal twist and focus on the tuning of the heavy doublet. We give an overview of the runs we have carried out and section3gives first results for some light-quark sector observables.
2. Lattice setup
In the gauge sector we use the Iwasaki gauge action [11]. With this gauge action we observe a smooth dependence of (possible) phase sensitive quantities on the hopping parameterκ around its critical valueκcrit. The fermionic action for the light doublet is given by:
Sl=a4
∑
x
{χ¯l(x) [DW[U] +m0,l+iµlγ5τ3]χl(x)}, (2.1) using the same notation as used in [10]. In the heavy sector, the action becomes:
Sh=a4
∑
x
{χ¯h(x) [DW[U] +m0,h+iµσγ5τ1+µδτ3]χh(x)}. (2.2) At maximal twist, physical observables are automaticallyO(a)improved without the need to de- termine any action or operator specific improvement coefficients. The gauge configurations are generated with a Polynomial Hybrid Monte Carlo (PHMC) updating algorithm [12–14].
2.1 Tuning action parameters
Tuning to maximal twist requires to set m0,l andm0,h equal to some proper estimate of the critical mass mcrit=mcrit(β) [8]. Here we setm0,l =m0,h≡1/(2κ)−4. As has been shown in [9], this is consistent withO(a)improvement defined by the maximal twist conditionamPCAC,l=0 (see also ref. [10]). The numerical precision at which the conditionmPCAC,l=0 is fulfilled in order to avoid residual largeO(a2)effects when the pion mass is decreased is, for the present range of lattice spacings,|ε/µl|.0.1, whereε is the deviation ofmPCAC,l from zero [4,15]. As explained in [10], tuning toκcrit was performed independently for eachµl value. From table1we observe that the estimate ofκcrit depends weakly on µl. The heavy doublet mass parameters µσ andµδ should be adjusted in order to reproduce the values of the renormalizedsandcquark masses. The latter are related toµσ andµδ via [8]:
(ms,c)R= 1
ZP(µσ∓ZP
ZSµδ), (2.3)
where the−sign corresponds to the strange and the+sign to the charm. In practice we fix the valuesµσ andµδ by requiring the resultingK- andD-meson masses to match experimental results.
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Label β κ aµl aµσ aµδ L/a T/a mπL |ε/µl|
C1 1.90 0.1632700 0.0040 0.150 0.190 20 48 3.0 0.14(14) C2 1.90 0.1632700 0.0040 0.150 0.190 24 48 3.5 0.07(14) A1 1.90 0.1632650 0.0060 0.150 0.190 24 48 4.1 0.03(3) A2 1.90 0.1632600 0.0080 0.150 0.190 24 48 4.8 0.02(2) A3D1 1.90 0.1632550 0.0100 0.150 0.190 24 48 5.3 0.02(2) A4 1.90 0.1632720 0.0030 0.150 0.190 32 64 4.0 0.08(7) A5C3 1.90 0.1632700 0.0040 0.150 0.190 32 64 4.5 0.04(5) A6 1.90 0.1632670 0.0050 0.150 0.190 32 64 5.0 0.05(2) D2 1.90 0.1632550 0.0100 0.150 0.197 24 48 5.3 0.35(1) B1 1.95 0.1612400 0.0025 0.135 0.170 32 64 3.4 0.06(6) B2 1.95 0.1612400 0.0035 0.135 0.170 32 64 4.0 0.02(2) B3 1.95 0.1612360 0.0055 0.135 0.170 32 64 5.0 0.08(1) B4 1.95 0.1612320 0.0075 0.135 0.170 32 64 5.8 0.05(1) B5 1.95 0.1612312 0.0085 0.135 0.170 24 48 4.6 0.01(2)
Table 1:Input parameters,mπLand|ε/µl|for all ensembles used in this paper. Every ensemble has∼5000 thermalized trajectories of length τ=1. We have two main ensemble sets: A and B, atβ =1.90 and β =1.95 respectively. Ensembles labeledCare used to check finite size effects. Ensembles labeledDare used to check/tune the strange and charm quark masses.
2.2 Determination of heavy-light meson masses
Since the twisted mass lattice Dirac operator of the non-degenerate heavy quark doublet (cf.
(2.2)) contains a parity odd and flavour non-diagonal Wilson term, parity as well as flavour are not anymore quantum numbers of the theory. In contrast to parity and flavour conserving lattice formulations, it is not possible to compute correlation functions restricted to a single parity and flavour sector in this setup. While theK-meson will remain the lightest state and therefore relatively easy to extract, for a theoretically clean determination of theD-meson mass one has to consider the four sectors labeled by parityP=±and flavour=s/cat the same time. And since besides the K-meson there are a number ofK+n×πstates and possibly also “positive parityKstates” below theD-meson, this renders theD a highly excited state. At currently available statistics it seems extremely difficult to extract such a high lying state.
As such, we resort to a different strategy in order to extract this mass. We attempt to determine the mass of theD-meson without computing the full low-lying spectrum, e.g. we do not determine all low lying states below theD. To this end we apply smearing techniques (cf. [16], where the same setup was used) to construct highly optimized trial states with large overlap to theK- andD- meson, and make certain assumptions about these trial states, which will be motivated and detailed in an upcoming publication. To extract theD-meson mass, we applied three different methods: (1) solving a generalized eigenvalue problem, (2) performing a multi-exponential fit and (3) rotating the twisted basis correlators back to the physical basis (in order to do this we need to compute the light and heavy twist angle and a ratio of the appropriate renormalization constants). The values of theD-meson mass extracted from these three different methods are consistent with each other.
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2.3 Status
The left panel of figure1shows the tuning of the strange quark mass by showing the difference, scaled with the chirally extrapolated value ofr0/abetween twice theK-meson mass squared and the pion mass squared. SetAat β =1.90, aµδ =0.190 (green points) appears to overshoot the physical point (the black cross on the left), while setB(red points) extrapolates better. To improve the tuning of the strange quark mass for setA, we are currently applying a reweighting procedure as described in [10] in the parameters aµδ and κ. The blue point with a different heavy sector splittingaµδ =0.197 is a run to check this procedure. Though this run is not tuned to maximal twist yet, theK-meson mass appears to be much closer to its physical value. The right panel of figure1shows the mass of theD-meson (obtained in this case by method (3)) as a function of the pion mass squared for various simulation points as well as the experimental value from the Particle Data Group [17] . The plot demonstrates that we have tuned the charm (sea) quark mass in our simulations to a physically realistic value. As a final check, we also use an estimate ofZP/ZS to verify thatmc∼10ms.
(r0mπ)2 (2mK2 −mπ2 )r02
0 1 2 3 4
●
● ● ● ●
●
0.0 0.5 1.0 1.5
(r0mπ)2
r0mD
0 1 2 3 4 5 6
● ● ● ● ●●
0.0 0.5 1.0 1.5
L/a
● 24 32 aµδ
● 0.17
● 0.19
● 0.197
Figure 1:r20(2m2K−m2π)andr0mDas functions of(r0mπ)2, showing the status of the tuning of the strange and charm quark mass respectively. The experimental value from PDG is added as the black cross (r0= 0.45(3)fm was used). Red points label theβ =1.95 runs, green points label theβ=1.90 runs, where the single blue point corresponds toβ =1.90 with a different heavy sector splittingaµδ. Circles denote runs withL/a=24, triangles indicate a volume withL/a=32.
3. Results
As a first check of our data, we have compared it to the extensively analysed data set that exists for ourNf =2 data. To compare the two sets, we plot dimensionless physical ratios in figure2. The figure shows no evidence of disagreement between all our results, suggesting small discretisation effects and small effects of dynamicals- andc-quarks for these observables.
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((mππ mN))2 ((mππfππ))2
2 4 6 8
●
●
●
●●
●
●
●
●
0.04 0.06 0.08 0.10 0.12 0.14
((r0mππ))2
r0fππ
0.0 0.1 0.2 0.3 0.4 0.5 0.6
●
●
● ● ●●
●
● ● ●
●
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 L/a
● 24 32 48 ββ
● 1.9
● 1.95
● 3.9
● 4.05
● 4.2
Figure 2: (mπ/fπ)2vs (mπ/mN)2 (left) andr0fπ vs (r0mπ)2 (right) for bothNf =2+1+1 data (with β =1.90,1.95) andNf =2 data (withβ =3.9,4.05,4.2 and using a different gauge action). In the left plot,mN is the nucleon mass, and the physical point is included as the black cross. In both plots finite size corrections are not applied, and in the right plot the chirally extrapolated values forr0/awere used.
3.1 Light meson chiral perturbation theory fits
In order to extract the lattice spacing and light quark mass from our data-sets, we perform a next to leading orderSU(2)chiral perturbation theory fit of themπand fπ data. We use continuum formulae and correct for finite size effects either without any new low energy constants à la Gasser and Leutwyler [18], or with ¯l1 and ¯l2 added in, as described in [19]. The results are listed in table2. We have performed these fits for ensemble setsAand Bseparately, and also combined them in a single fit. In table2, we include a systematic error, estimated at 2−5%, coming from the dispersion of the values of the fitted parameters between NLO and NNLO. Note that since the quark mass enters theχPT expression, in order to combine the two sets at different lattice spacings, we need to know the renormalization factor of the quark massZµ=1/ZP, a computation which is not yet complete. Assuming thatZPis effectively a function ofβin the range of parameters we are considering, we can fit the ratio of thoseZP-values and lattice spacings and extract lattice spacings from the combined fit. In every fit we use as inputs the physical fπ andmπ, and extract f0, ¯l3and l¯4. A complete analysis (analogous to [20]) of the systematic effects is in progress.
set pts f0(MeV) l¯3 l¯4 aβ=1.90(fm) aβ=1.95(fm) A&B 11 121(4) 3.5(2) 4.7(2) 0.086(6) 0.078(6)
A 6 121(4) 3.4(2) 4.8(2) 0.086(7)
B 5 121(4) 3.7(2) 4.7(2) 0.078(7)
Table 2:Results from the NLOSU(2)χPT fits for combined, only setAand only setBrespectively. Errors are dominated by a systematic error of 2−5% due to performing an NLO fit. The column "pts" refers to the number of ensembles used in that fit.
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3.2 Chiral extrapolation of the nucleon mass
In this section, we present preliminary results for the light quark mass dependence of the nucleon mass. We consider the one-loop result from heavy baryon chiral perturbation theory (HBχPT)
mN=m0N−4c1m2π− 3g2A
16πfπ2m3π (3.1)
and fix the scale and light quark mass to the point where the ratiomN/mπattains its physical value.
We fix fπ and gA to their physical values (130.7 MeV and 1.27 respectively) as has also been performed in [7]. Using this procedure, we find a lattice spacing of 0.089(2) fm and 0.077(3) fm forβ =1.90 and 1.95 respectively. The χ2/(d.o.f.)of these fits is not very good, and fitting a linear extrapolation appears to be consistent with the data. This is not unique to our data, and has been observed by various collaborations. We therefore perform the linear fit here as well, and absorb the difference between the two extrapolations in the systematic error. A more detailed analysis of the chiral extrapolation of the nucleon mass will be presented in an upcoming study.
The lattice spacings that we obtain from the chiral extrapolation of the nucleon mass are 0.089(9) fm and 0.077(4)fm for setAatβ =1.90 and setBatβ =1.95 respectively.
3.3 r0/aextrapolation
Since r0/ais very sensitive to κ in the vicinity ofκcrit, the fact that we now tune to maxi- mal twist at every value ofµl, might, with respect to what was done for theNf =2 case, in part provide an explanation for the observed change of slope in the mass dependence ofr0/abetween Nf =2 andNf =2+1+1. Note however that these differences tend to diminish when increas- ing the value in β in theNf =2+1+1 case. We extrapolate r0/a using a simple quadratic fit r0/a=c1+c2a2µl2, wherec1 is the value ofr0/ain the chiral limit. We perform both a polyno- mial fitr0/a=c1+c2aµl+c3a2µl2 and a linear fitr0/a=c1+c2aµl to help estimate systematic errors. We find that based on theχ2/d.o.f. the quadratic fit is for both values ofβ the best fit. The polynomial fit gives nearly identical results for c1, whilec1 from the linear fit is 1 to 3σ higher.
Using the lattice spacings from the combined light meson chiral perturbation theory fit, we extract two predictions forr0, which seem to agree well atr0=0.45(3)fm.
β c1(quadratic fit) a(fm) r0(fm) 1.90 5.24(2) 0.086(6) 0.45(3) 1.95 5.71(4) 0.078(6) 0.45(3)
Table 3: r0determination for both ensembles separately. c1(qua) is the value of a quadraticr0/aextrap- olation in the chiral limit with the statistical error in brackets. The lattice spacingsa are taken from the combined light meson chiral perturbation theory fit. The obtained values forr0from the two ensembles seem to agree well with each other.
4. Conclusions
We have presented first results from runs performed withNf =2+1+1 flavours of dynamical twisted mass fermions. No evidence of disagreement between these results and those withNf =2
PoS(LAT2009)104
twisted mass fermions is shown through dimensionless ratio plots (ofmπ, fπ,mN andr0/a), sug- gesting small discretisation effects and small effects of dynamicals- andc-quarks for these observ- ables. We have extracted the lattice spacings of our two ensemble sets using two different methods, which agree within errors with each other. We have measuredr0 on both ensembles and found consistent results. We are in the process of performing a more detailed combined analysis in order to improve our understanding of the systematic errors.
Acknowledgments
We thank all other members of the ETM Collaboration for valuable discussions. The HPC re- sources for this project have been made available by the computer centres of Barcelona, Groningen, Jülich, Lyon, Munich, Paris and Rome (apeNEXT), which we thank for enabling us to perform this work. This work has also been supported in part by the DFG Sonderforschungsbereich/Transregio SFB/TR9-03, and by GENCI (IDRIS - CINES), Grant 2009-052271.
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